3:2 spin-orbit resonance in Mercury. How it works? Why It didn't fall in a 1:1 spin-orbit resonance? Why Mercury is not tidally heated having an eccentric orbit and not been tidally locked? Where does Mercury's orbital eccentricity comes from?

2:3 mean motion resonance between Pluto and Neptune. How it works and why is it stable? Plutinos.

Other families of resonant TNOs

2:4 mean motion resonance between Tethys and Mimas (and the libration of the nodes).

3:4 mean motion resonance between Hyperion and Titan.

TRAPPIST-1 complex chain of 6 mean motion resonances.

Titius-Bode Law and Kepler's Mysterium Cosmographicum in solar and extrasolar systems as an indicator of to stable resonances.

I'm now finishing my Physics career and have been working for a year at ESA's facilities in Madrid, and after all of that I'm still fighting to grasp some understanding of celestial mechanics, an interesting topic (that it's starting to obsesses me) with profound consequences. My personal motivation is that even if I'm learning a lot I lack the most basic intuition for this part of astrophysics and would like to learn from you (ahem, Watsisname please), very smart and educated people that I know I can learn so much from.

But there's another reason for this thread (one less egoistical) and that is that in the process of learning some of the topics above I've encountered bad explanations (even at University level), filled with terrible misconceptions that I've been fighting against a lot. I would like to tell you some of them as to avoid more and more confusion in these topics, at least for the ones that find the information useful.

GOALS (EFFECT)

To gain an in-depth understanding on celestial mechanics in the Solar System. To gain intuition on that understanding and not make it just by "the calculations say so even If I'm unable to comprehend what is going on". To educate ourselves and improve SpaceEngine with accurate suggestions on the matter.

RULES (LAWS OF CAUSALITY)

I'm going to try to address some topics when I can and explain them the best way I can, but the vast majority of them are to far away from my understanding so I'm also going to ask a lot of questions. I hope to be more or less constant with this (but I would need time). I'm also planning to do some Python programs to try and test some ideas that I have (probably also misconceptions) about how really Laplace resonances work for example. But my main goal here is to have a "low-level" discussion with all the necessary math but above all with real comprehension of what is going on. For that I think is important to use analogies. Let's make a very very difficult topic part of popular science, step by step, for everyone to understand even if the process is long and slow.

You can (and should) also try to address each of the topics. The order is not important.

If we address a topic I'm going to link where we discussed in the index of topic of the main post (this one). If a new topic should be addressed or one of the topics needs to be splittled in two then we can generate a larger index.

I will start soon with the tides (since I think I finally have overcomed some of the largest obstacles in understanding them). See you then.

You chose the coolest and most appropriate image possible to head this topic. Great idea, and this should be very interesting and fun to work through. I have no clue how much we can cover or how in depth we can go, but I'm excited for it!

I am currently writing up some explanation for the last one (precession of Mercury's perihelion), but it might be a little while before I finish since I have many other things to do under time constraints. So if anyone was thinking of discussing that one, please hold off because I would like to cover it in detail.

Often in discussions of Mercury's perihelion advance, the effect is shown greatly exaggerated. Sometimes the orbit is even shown much more eccentric than it really is. Before we dive in, let's take a moment to get some perspective on the true shape of the orbit and appreciate the scale of the solar system.

The Solar System (June 12, 2018)

All orbits are ellipses, but most planetary orbits are close enough to circular that it can be hard to notice the eccentricity at a glance. Mercury is the biggest exception with an eccentricity of 0.21. (Or Pluto at 0.25). What's interesting, and what this post is all about, is that Mercury's orbit is not stationary. The ellipse slowly shifts around, its perihelion point advancing by about 574 arcseconds (or 0.159 degrees) per century. Much of this (531 arcseconds) can be explained by Mercury being perturbed by the other planets. However, the remaining 43 arcseconds per century are "anomalous", unaccounted for by Newtonian mechanics.

The puzzle of this additional observed precession was first noted by Le Verrier in 1859. Following the successful prediction of the existence and location of Neptune from similar perturbations on Uranus' orbit, Le Verrier proposed that Mercury's orbital precession could be caused by another undiscovered planet inside of Mercury's orbit, which he named Vulcan. Astronomers searched for this missing inner planet during solar eclipses. Of course, Vulcan was never found...

The mystery was not solved until Einstein began developing his general theory of relativity. When he applied his general relativistic equations to the problem in 1915, he found that they exactly predicted the additional 43 arcseconds per century! He considered this one of his greatest achievements.

Imagine my joy at the recognition of the feasibility of general covariance and at the result that the equations correctly yield the perihelion motion of Mercury. I was beside myself for several days in joyous excitement.--Albert Einstein

Physics:

So, why does it happen? Is there an extra energy involved? (Spoiler: Yes, in a way!)

For an elliptical orbit, we can think of the orbital motion as being made of two parts: an oscillation in azimuth (angle around the Sun), and an oscillation radially (in and out). In Newtonian mechanics, the periods of these two oscillations are exactly equal. That is, each time the planet completes one cycle around the Sun, it also exactly completes one cycle radially in and out. Therefore in Newtonian mechanics there is no precession of the orbit (besides that which is caused by the influences of the other planets).

In general relativity this is no longer true. The periods of the two oscillations are different! To see why, it is easiest to examine the shape of the "effective potential", which you can think of as being just like the potential energy well around a spherical mass, except also taking into account the angular momentum of the orbiting body.

If you are unfamiliar with the concept of a potential energy well, imagine a rolling landscape of hills and valleys. If you place a ball at the bottom of a valley, it will just sit there. It's in a stable equilibrium there. On the other hand, the top of a hill is an unstable equilibrium. If you displace the ball slightly from the top of the hill and let it go, it will continue to roll down, exchanging gravitational potential energy for kinetic energy as it drops. Finally, if you have some bowl-shaped valley and release the ball from rest somewhere on the slopes, then it will roll down to the bottom, and (ignoring friction) keep going, rolling back up the other side until it reaches its former height, and then roll back again, oscillating back and forth indefinitely.

This modelling of potential energy is extremely useful. On Earth's surface for example, the gravitational potential energy of an object is simply proportional to its altitude, and therefore the shape of the potential is exactly the same as the shape of the landscape. But we can apply potentials to many other situations. In general, we can use "potential energy wells" as a way to understand motions of objects subjected to different kinds of attractive and repulsive forces, from balls rolling down hills, to the vibrations of atoms bound together in molecules.

An elliptical orbit is an oscillation in an effective potential well, as well. There is a hill to climb as you move outward due to the gravitational attraction of the central mass, and there is also a hill to climb as you move inward! This is because your sideways motion is associated with angular momentum, which produces a repulsive centrifugal force. By conservation of angular momentum, the sideways velocity increases as your orbit swings inward, increasing the centrifugal repulsion and making it harder to approach the center. If the angular momentum is large enough, then the inward fall can be halted, and you swing back out again, making an orbit!

So if we can compute the effective potential well for a planet orbiting the Sun, we can figure out some things about its motion. The anomalous precession of an orbit can also be understood this way, by comparing the motion from the Newtonian potential with what happens in the general relativistic version of the potential. Let's try it!

Math:

The goal here is to try to understand how things work conceptually, and I will try to emphasize those concepts (especially physical concepts) as we go along. The hard truth is that much of it will still require going through the math in order to access it, but I'll do my best to turn that math into something visualizable and comprehensible.

In Newtonian mechanics, the gravitational potential energy of a mass m a distance r around a spherical mass M is

This describes a simple curve which plummets downward indefinitely as you move to smaller radii. As it should. If you drop something near a massive object and give it no sideways motion (no angular momentum), then it will fall straight into it.

If we instead give the object some angular momentum L, then the potential is modified:

where μ is the reduced mass, which for a planet orbiting the Sun (m << M), reduces to approximately m.

Let's see what this looks like for Mercury:

Again the way to think about this is that the potential describes a landscape that a ball will frictionlessly roll across. To visualize that, I added the large red dot to represent Mercury's position as a function of time. The acceleration is determined by the slope of the potential, and I plotted the motion with 2 days per frame. With 44 frames in all, this traces out one complete period of Mercury's orbit in the radial motion. I also added a horizontal line to show the total energy of Mercury (its gravitational potential energy + kinetic), which is constant along the orbit. Where this line is above the effective potential defines the range of distances from the Sun that Mercury's orbit will cover. The vertical lines represent the extremes (Mercury's perihelion and aphelion distances).

Why is the Newtonian effective potential shaped this way?Because of Mercury's amount of angular momentum, the shape of the effective potential that it sees near the Sun is a valley with hills on either side. The hill at large radii is due to the Sun's gravitational attraction, while the hill at small radii is due to centrifugal repulsion. Even though the gravitational force grows stronger at closer distances, for an orbit the centrifugal force gets stronger more quickly, due to the conservation of angular momentum which increases your sideways speed.

A useful concept to keep in mind here is that the period of the radial oscillation depends on the curvature of the potential well. If the well opens up more sharply, then the average acceleration is greater, and the period is shorter.

And here's one other useful trick. If the amplitude of the oscillation is not too large (does not reach too far away from the minimum of the well), then we can approximate the well as a parabola around the minimum. Then for a parabolic well the motion is described very simply as a simple harmonic oscillator. For Mercury this approximation isn't particularly good (the well is noticeably asymmetric over the region Mercury covers), but it's not terrible either.

For a simple harmonic oscillator, the frequency is given by

which for the radial motion leads to

The frequency for the angular oscillation on the other hand is given by

Which leads to the exact same expression:

So we see in Newtonian mechanics the periods are exactly equal and there is no anomalous precession. Now let's see how this gets modified when we move to General Relativity.

The Effective Potential in General Relativity:

Einstein's general theory of relativity describes gravitation as distortion of the geometry of space-time. I'm sure you've heard of the rubber-space-time sheet analogy, and perhaps seen the interactive displays at science museums where you can roll coins or marbles down a funnel. Something like this (I love this guy's presentation by the way).

These are classic and excellent tools for teaching how mass distorts the shape of space-time, and how the shape of space-time gives the orders for how other masses will move. Now if you watched it carefully you might have noticed something interesting. Not only can you get elliptical orbits in these demonstrations, but highly elliptical orbits that fall deep down into the well also precess!

We could just stop right here and say "this demo explains Mercury's precession"! But that would be not quite right. The reason precessing orbits appear in these funnels is because their shapes do not match the Newtonian potential, and in general if you change the shape of the potential you can make all kinds of weird trajectories occur. But these funnels also do not correctly reproduce the general relativistic potential. So the motions we see on them do not correspond to real celestial motions, even if they appear qualitatively similar.

A motivation of your post was to go beyond the common but not completely correct explanations to get closer to "what's really going on". So let's get the motions "the right way". We will use the general relativistic effective potential for an orbiting body:

Again I don't want to get lost in math, but it's worthwhile just to look briefly at what the math is saying here. Notice this still has the exact same two terms from the Newtonian effective potential: an attraction that goes as -1/r, and a repulsion that goes as +1/r2. But a new term is added: another attractive term that goes as -1/r3. This means that at very small radii, the -1/r3 term dominates, and gravitation becomes attractive again, dominating even over the centrifugal effect of your orbital velocity.

Next we will apply this to Mercury.

Mercury's Orbit in General Relativity:

Here's where everything comes together. Now we can gain some insight by plotting Mercury in the general relativistic potential. We have one small hurdle though. The Sun's gravitational field is pretty weak by general relativistic standards. If I plot the general relativistic potential on top of the Newtonian potential, you will not be able to see the difference between them.

I guess I could just plot the difference between the two... but I have a better idea. I'll instead "make general relativity stronger", by reducing the speed of light to 1/1000th of its actual value. Here's what happens:

Observations:

The minimum in the effective potential well is deeper and displaced slightly inward.

The oscillation spreads over a wider range of radii for a given energy than before.

The well opens up a little more steeply.

Remember that for oscillations that are not too large about the minimum of a well, the frequency of the oscillation is related to the curvature. Because this well opens up more steeply, we should expect Mercury's radial oscillation to be a bit faster than before. To check, I iterated through the radial and angular equations of motion and plotted the results, for the Newtonian case and for general relativity with c slowed by a factor of 1000. The radial motion is the black curves while the angular motion is in blue. Vertical blue lines represent the completion of one 360° circulation about the Sun, while each peak in the black curves represent one complete oscillation radially (from aphelion to aphelion).

Indeed, with general relativity "turned on", the radial oscillation is faster than before. But so is the angular oscillation, even more so! The two oscillation periods are unequal, and Mercury completes one 360° revolution in less time than it takes to complete one oscillation radially. Now at last we directly see the precession! Here it is as an orbital plot:

Why is the time to complete one angular orbit reduced so much, and more so than the radial one?

Here we have a well which has moved slightly inward. With General relativity turned on, Mercury plunges inward a little closer to the Sun, where by conservation of angular momentum, it circulates a little bit faster. Furthermore, the shape of the potential has changed. Because of the extra -1/r3 term added by general relativity, the potential is slightly less steep near perihelion. There Mercury experiences slightly less acceleration in the radial direction, and is slower to swing back outward from perihelion. It hangs around and circulates there a little bit longer. Therefore by the time it moves back out to aphelion to complete one radial oscillation, it has completed more than one angular oscillation, and its orbit precesses.

It all ultimately arises from the change in the effective potential, and to call back to the question of whether energy is involved, an effective potential defines the change in potential and kinetic energy as an object moves through some landscape (curved space-time in this case). So yes, this orbital precession can be thought of as an effect of how general relativity modifies the exchange of potential energy, by changing the geometry of space-time.

I should also say that this anomalous precesion does not only happen for Mercury. It happens to all orbits! But it is strongest for Mercury, since it is closest to the Sun. These are values (arcseconds per century) for all the inner planets:

Mercury: 42.98

Venus: 8.62

Earth: 3.85

Mars: 1.35

That, I think, completes the story of Mercury's Perihelion Advance. We've seen how to model motions by using effective potentials, and applied them to Mercury to visualize the changes that arise from general relativity. By "turning up general relativity", we can see the precession in even a few orbits and understand why it happens.

However, there is still more that we can cover on this topic. The Sun's gravitational field is weak, so we've only explored the weak-field effects introduced by general relativity to orbital motions. Really amazing things happen if we move into stronger fields! For anyone interested, in the next section I move away from the solar system, and explore what happens to orbits near black holes.

Going Further: The Bizarre Orbits in Strongly Curved Space-Time

Recall back to the expression for the general relativistic effective potential, and the attractive -1/r3 term it introduced. At very small radii, this term will dominate, even over the centrifugal effect caused by angular momentum. Instead of a hill at small radii, we get a hole. This is exactly why there are black holes. Get too close to a sufficiently massive and compact object, and the gravitation totally overwhelms. Even light cannot move fast enough to withstand it.

The Sun is very far from being a black hole. To become one, it would have to be squeezed down into a space smaller than 6km across. Whereas Mercury orbits at around 60 million kilometers away. So Mercury doesn't really explore this region of strong general relativistic effects. But, as I was working on making the plots for this post, I suddenly realized "I've just made something that simulates orbits in general relativity."

Let's change some parameters, and look at some orbits that happen very close to a black hole. Don't worry, there is no more math. I'll just have some neat figures and explanations.

The effective potential near a black hole:

Here I've plotted the potential for a particle with some angular momentum, and starting at 10 event horizon radii from the black hole. The mass of black hole I used is 4x106 solar masses which is similar to the supermassive black hole at the center of our galaxy. The energy of this particle is just below that of the hill on the left side, so that its orbit can drop down close to the black hole without falling in.

(Aside: The units 'r/M' for the horizontal axis means that I'm plotting distance in terms of units GM/c2. In these units, r/M=2 represents the event horizon. r/M = 3 defines the photon sphere where light can orbit around the hole, and r/M=6 defines the "ISCO" or the innermost stable circular orbit that can exist around the hole.)

Because the particle will slow down a lot near that peak, you might imagine it could spend a bit of time circulating there at that radius before moving back out. Sure enough, it does.

This trajectory is delicately balanced on a knife's edge. If it strayed just a little bit further in, it would plunge down the other side, into the black hole. Like so:

So this is one remarkable feature of motions near a black hole. You can get an orbit that drops down and then circulates around several times very close to the black hole, before either zooming back outward to safety, or fatally falling down in. And the difference between the two is precariously thin.

Another remarkable thing can happen here. You actually can get orbits that retrace themselves -- but they will be very different from ellipses. With the right angular momentum and energy, the orbits become beautiful flowery patterns. There is absolutely nothing like these trajectories in Newtonian gravity.

There is a wonderful "Periodic Table" of black hole orbits, including ones for rotating (Kerr) black holes. Sadly, as beautiful as these are, they probably do not occur in nature. Not just because meeting the initial conditions for them would be unlikely, but also because for real objects in these orbits, gravitational radiation (gravitational waves) would be significant and cause them to decay and change shape fairly quickly. However, it is faintly possible that we could sometime observe something at least briefly resembling one of these orbits, in the gravitational waves emitted by a binary black hole merger. And that would be pretty amazing to see.

Well, shoot. I wanted to split that up into spoilerized sections to make it less obnoxious to read, but spoiler tags seem to break with it.

P.S.The length of that post is not indicative of what I expect everything else will or should look like. It's just that relativity is one of my favorite things, and working through the details became a labor of love. Expect that I will occasionally drop in new posts to cover other things in the list in more digestible segments.

I hope that others will also contribute! This thread should not be just me and FFT.

Some complimentsI would like to quote each paragraph and say things like "Awesome!", "amazing", "wow, never thought of that" and "very well explained" but since that would just be hard to read and redundant I'm just going to throw a global "SPECTACULAR". Good job Watsisname. This is the first time I read a clear explanation of the phenomenon. Very very interesting.

Also, good job at showing what happends when the potential gets more and more terms. I loved the periodic table of black hole orbits too.I found a youtube video that tries to explain the same but gets way behind you in terms of making it clear. It's interesing still because of the connection between orbits around black holes and light paths and the distortion of background images by gravitational lensing, so I'll leave it here

This is exactly the kind of explanations I was looking for. A bit of math to make all consistent, many examples, and step by step intuitive aproach. Thank you.

Some questions/clarificationsA) So let's see if I've understanded how energy works in both models. The first time I heard about the precession of Mercury's perihelion I thought there was some external energy been injected in the system to perform that movement. In fact, as you explained, the vast majority of Mercury's perihelion precession is accounted by the gravitational tug of the other planets (there is a transfer of energy from them to Mercury so that the orbit changes). So, without a detailed explanation given to me about the relativistic effect, I thought that some kind of energy was been added or subtracted from the system by ... well, space-time itself (I just supposed that gravitational waves could be invoked in some exotic way). So, totally wrong and acritical about what I thought.

It's interesting because you gave more and more examples of scenarios were this "extra energy" seems to be needed. The fact that one could be orbiting a black hole and suddenly get ejected into a larger orbit around seems totally conterintuitive if energy is to be conserved and the system is isolated from any external influence. But as I now comprehend, this is just perceived and there is no extra energy been pumped into the system, just the fact that we think "Newtonianly" and that strange behaviour would need to invoke some extra energy in a newtonian framework.

I can see clearly why there's an illusory effect now using one of your diagrams.

For the same total energy (cyan line) a "Relativistic Mercury" would arrive to higher aphelia than a "Newtonian Mercury". For a Newtonian mind that would be seen as if Mercury had gained more potential energy (sky-blue line) to arrive at those heights, from somewhere. The effect is even stronger for perihelia; for the same total energy a "Relativistic Mercury" would arrive to much lower perihelia than a "Newtonian Mercury" meaning, from a Newtonian perspective, that Mercury is now moving with higher kinetic energy than expected (as if the Sun got a stronger gravitational pull than expected supontaneusly). It would look like Mercury had gained a lot of energy from this (dark blue line). But all of this is wrong because Mercury doesn't follow a Newtonian effective potential but a relativistic one and therefore our Newtonian expectations are wrong. Energy is conserved even if that motion would make us (intuitive newtonians) fall in the erroneous assertion that energy is coming and going from the system.

Also very interesting to note that even if we wanted our "Newtonian Mercury" and "Relativistic Mercury" to have the same aphelia 1) we would need the "Relativistic Mercury" to have less total energy and 2) even then the perihelion of the "Relativistic Mercury" would trick us into thinking that it has much more energy than we would expect for a "Newtonian Mercury". I find that alone fascinating; A system that actually has less energy in a relativistic framework seems to behave as it had much more energy from a Newtonian perspective.

I suppose that for the precession of the perihelion something similar happens. Is not that there is some mysterious energy driving this orbital change, is just that in reality, in the relativistic world we live in, there are no Keplerian orbits.

B) As another idea for the debate: What is the connection of this phenomenon and gravitational time-dilation? Or it's even related in some way? The fact that time flows differently in different parts of Mercury's orbit can be interpreted as the other side of the coin of this explanation?

C) I'm also fascinated by the fact there are two independent frequencies at play here. So a Keplerian orbit is just a 1:1 frequency ratio between radial motion and angular motion, right?. And those flower-patterns around black holes would also be ratios between small integers. For example the 4-petal would be a 1:4 ratio between both frequencies. So, those orbits have very special properties, since they are all closed repeating paths (something that wouldn't happend for non simple ratios) and their aphelia repeats the same location every certain time. I'm wondering if that could create interesting mean-motion to relativistic-perihelia-precession resonance in particles in the disk of a black hole. Is there anything published about this?

D) I thik I would like to know more about the additional term for the generalised potential introduced by relativity. Where it comes from? I'm not feeling complete without this haha. I know it probably involves much more complex math (I don't really remember having seen this in class).

E) The "where is the energy?" debate leds to the realisation that gravitational forces in relativity would be non-conservative as oposed to gravity in Newtonian Mechanics. But still the stress–energy–momentum pseudotensor it's conserved. I would love to hear an explanation as well written as yours for something like that.

Some ideas for the general threadA) Whatsisname, If you are willing to do so, I think we would all like to hear about relativity in the Solar System. I couldn't come up with any other topic in the index but I'm sure you would have many ideas. Tell me so I can add that to the list of topics we could adress here.

B) For example, now that you talked about this, I'm interested in sungrazers like C/2011 W3 (Lovejoy), that have extremely elliptical orbits with extremely low perihelia. What relativistic effects do they have to deal with? and what consequences does it has? I immagine the precession of perihelia of comet Lovejoy to be extreme in comparison with Mercury's.

Comet Lovejoy got to just 140,000 km from the surface of the Sun on the 16th of December 2011, which means that at that time it was 140,000 km + Sun's Radius = 140,000 km + 695,700 km = 835,700 km = 0.0056 AU from the center of the Sun. If my calculations are correct, that would mean that the hour (measured from our perspective) it spent there (or nearby) would have been 6.3 seconds shorter in the comet, just by gravitational time dilation, and not even a hundreth of a second due to time dilation from the 536 km/s speed the comet archived at perihelion (0.18% of c). So comet Lovejoy actually travelled around 6 seconds in time! A timespan that would be noticiable for a human been.

So considering that relativity plays important roles, I suppose that no comet's Lovejoy orbit looks the same in a diagram right?

I also hope to see your explanations, SE community!. Mine about tides would have to wait a little more since it's a crazy mess in which even people like Brian Cox, Neil de Grasse Tyson or even organizations like the NOAA make popular explanations that are totally wrong and misleading, so give me some time since I want to make sure my explanations fit well within the actual scientific consensus and I have enought examples and analogies to make it easy to understand.

Watsisname, very good job in explaining the Mercury orbit! I have a question though: is time/space dilation due to Mercury's velocity accounted in equations automatically, or its effect is simply negligible?

Your post deserves not even its own thread, but a publication as an article! Maybe even on SE website? I can add a new section, similar to the blog, where we can make a posts on astronomy and science. What do you think? There are many interesting questions and perfect explanations lurking in this and old forum. If someone indexing them, we could convert them into articles for the new website pages.

Wow, thank you SpaceEngineer, and I'd be honored! I think that, especially as more great explanations accumulate in this thread and combined with the Astronomy Q&A thread, it would be great to have something like a section of the website dedicated to them as an educational tool.

I have a bit of time to go over some of your questions and hopefully can address most of it:

SpaceEngineer wrote:

Source of the post is time/space dilation due to Mercury's velocity accounted in equations automatically, or its effect is simply negligible?

Both, in Mercury's case. The equations are expressed in terms of the time measured by an observer fixed to Mercury (the proper time), so it does account for time dilation. But the effect makes little difference for computing Mercury's precession anyway.

The choice of "whose time go by" does become important for the black hole examples. If we went by time measured by an observer far away, then the trajectory of an object dropped into a black hole would slow down and stop at the horizon. But from its perspective, it does cross the horizon and meets the singularity quite quickly.

Since the equations used go by proper time, they show the trajectory extending through the horizon as the object itself would report. This also means the "radial and angular motions" plots correctly relate what the object reports as its position as a function of the elapsed time that it measures. Those ~20 minutes of "hang time" near the black hole in the nearly-plunging orbit are in fact 20 minutes as measured by someone making that trip. Another prediction is that if you fell into the black hole Sagittarius A*, starting from rest very far away, then after crossing the event horizon it would take just 28 seconds to hit the singularity.

Yes! You got it perfectly! I'm glad you were able to clear up the earlier confusion, and explain it back so clearly with these extra insights.

B) (Time dilation's role)As described above, this analysis is done in terms of proper time for the object on the trajectory. I would not be surprised if in some way the precession could be explained in terms of time dilation, but I don't think it would be very straightforward, if it works at all. For example, gravitational time dilation affects both the radial and tangential motions equally, so there has to be more to it than that.

The curvature of space would also be something to consider. It might even be more directly related, as it does affect the radial and tangential motions differently (radial motion near a massive object is slowed down more). This causes the Shapiro Delay, which discuss in the Q&A thread here, and I also talk a bit about the spatial curvature here. But, I don't know offhand if this effect is exactly equivalent to what causes the precession or not.

C)

I'm also fascinated by the fact there are two independent frequencies at play here. So a Keplerian orbit is just a 1:1 frequency ratio between radial motion and angular motion, right?.

Yes, that's right. We're so used to Keplerian/Newtonian orbits that it can be a bit surprising to encounter this idea that orbital motion actually has two parts and that they can be different in frequency.

In the case of the last example with the 4-petal orbit, the ratio is actually 5:4. The orbit complete five 360° circulations in the same time that it completes four complete radial oscillations from aphelion to aphelion. As for whether this might happen in nature with accretion disks, I think probably no. Particles in an accretion disk make poor approximations of free particles, as friction and drag effects become important. Motions in disks instead become fairly circular, or spiral down in to the hole. Magnetic fields can also be important to the disk structure.

D)

I think I would like to know more about the additional term for the generalised potential introduced by relativity. Where it comes from?

Oh man, I should have explained that! I described briefly where the 2nd term in the Newtonian effective potential comes from and how it makes physical sense, but not for the 3rd term in GR. Yes, we can make some sense of where it comes from! (This will be more of a heuristic argument than a rigorous derivation, which would probably require some high level math working from the field equations.)

In Newtonian gravity, the sideways speed of an object on an elliptical trajectory increases at smaller distances, in order to conserve angular momentum.

The same thing is also true in general relativity. But it can only be true up to a point. The speed cannot exceed the speed of light! So there is a limit to how close an orbit can get -- below that point it simply cannot move quickly enough to generate the required amount of centrifugal force to keep it in orbit. What this means is that in general relativity, angular momentum becomes less and less effective at generating a repulsive centrifugal effect at closer distances. Thus the third, negative (attractive) term in the general relativistic effective potential.

Think also about how the speed of light appears in the expression. If the speed of light was infinite, then you could go arbitrarily fast to maintain an orbit arbitrarily close to a singularity. The third term in the effective potential would vanish and you recover the Newtonian gravity, which doesn't consider the finite speed of light! But the more you decrease the speed of light, the less you are able to orbit close in. The whole effective potential drops down, and more so at closer distances. This is why the effective potential for black holes has a steeply plummeting region near the event horizon. And for Mercury orbiting the Sun, the potential minimum gets lower and closer, and the well more steep.

E)

The "where is the energy?" debate leds to the realisation that gravitational forces in relativity would be non-conservative as oposed to gravity in Newtonian Mechanics.

The gravitational force is still conservative in this case. Only if a force is conservative can a potential energy be defined for it. This means that for any closed path through the space the net work done is zero and there is no net change in your energy. Otherwise, your potential energy at a given point in space would be ambiguous, and depend on the path that you take.

An example of a non-conservative force is friction, since there is always a loss in kinetic energy the more you move with friction being applied, and you never get that lost energy back.

General relativity does open some avenues for energy to not be conserved though. If the space-time is expanding or contracting then energy conservation itself becomes a tricky subject. For example, as the universe expands, the number of photons per unit volume decreases. Which is fine -- that doesn't violate any intuitions. But the energy per photon also decreases since the photons also get redshifted. So it seems like the energy of radiation in an expanding universe is not conserved. Similarly for dark energy, whose energy density remains constant as it expands. This counter-intuitive behavior is related to Noether's Theorem, for which PBS Space-Time has a nice episode on.

Great questions, and thanks again for the feedback!

Added:

FastFourierTransform wrote:

Source of the post For example, now that you talked about this, I'm interested in sungrazers like C/2011 W3 (Lovejoy), that have extremely elliptical orbits with extremely low perihelia.

This would be interesting to investigate. Unfortunately my current code cannot handle it -- to get the required accuracy it needs very short time steps near perihelion, but then that absolutely explodes the computation time for the rest of the orbit. It would take something like 2 weeks to compute it!

Probably I could solve this by make the time step variable and longer at larger distances, but I'd need to sit down and think about the best way to implement that.

Source of the post Probably I could solve this by make the time step variable and longer at larger distances, but I'd need to sit down and think about the best way to implement that

Don't worry Watsisname you made such a great work with all your explanation. No need to see the actual orbit. I was just curious about it. Very well done.

Watsisname wrote:

Source of the post Otherwise, your potential energy at a given point in space would be ambiguous, and depend on the path that you take.

Ops! Yeah that totally makes sense. Stupid question. Sorry

Watsisname wrote:

Source of the post What this means is that in general relativity, angular momentum becomes less and less effective at generating a repulsive centrifugal effect at closer distances.

That "heuristic explanation" is exactly the one I was hoping for With this I think anyone can read an intuitive and educational aproach. The mathematical derivation is a logical chain that starts with Einstein's postulates and the main theorems of differential geometry that surely would be interesting to read, but this explanation brings intuitive association of ideas in the narrative so it's more educational and one feels the sense of it all.

Watsisname wrote:

Source of the post I would not be surprised if in some way the precession could be explained in terms of time dilation, but I don't think it would be very straightforward, if it works at all. For example, gravitational time dilation affects both the radial and tangential motions equally, so there has to be more to it than that.

That looks like an extremely interesting topic I think. It would be like explaining the same phenomenon with special relativity instead of general. The idea that Mercury's "trajectory throught time" could be responsible (in some way) of that precession intrigues me now.

Source of the post Don't worry Watsisname you made such a great work with all your explanation. No need to see the actual orbit. I was just curious about it.

Well I wanted to find out what it would do as well, plus I'd like to be able to handle more general scenarios. C/2011 W3 Lovejoy is about as extreme as it gets in our solar system with an eccentricity of 0.99993 and coming within 1.2 radii of the Sun!

Now I have the code do variable time steps, such that the distance moved in each step is always some set fraction (typically less than 10-6) of the distance from whatever it's orbiting, which turned what would be a several weeks long computation for the comet into something just over an hour. I ran it a few times and as I turn up the precision the solution appears to converge to around 3.5 arcseconds per orbit! Much more than Mercury (about 0.1 arcsecond per orbit), but still so small that you wouldn't really be able to see it in an orbital overview, and I imagine other effects like outgassing would mask it.

Source of the post Don't worry Watsisname you made such a great work with all your explanation. No need to see the actual orbit. I was just curious about it.

Well I wanted to find out what it would do as well, plus I'd like to be able to handle more general scenarios. C/2011 W3 Lovejoy is about as extreme as it gets in our solar system with an eccentricity of 0.99993 and coming within 1.2 radii of the Sun!

Now I have the code do variable time steps, such that the distance moved in each step is always some set fraction (typically less than 10-6) of the distance from whatever it's orbiting, which turned what would be a several weeks long computation for the comet into something just over an hour. I ran it a few times and as I turn up the precision the solution appears to converge to around 3.5 arcseconds per orbit! Much more than Mercury (about 0.1 arcsecond per orbit), but still so small that you wouldn't really be able to see it in an orbital overview, and I imagine other effects like outgassing would mask it.

Lovejoy has the most eccentricity? Because in SpaceEngine there is a comet that moves way farther away from the sun, with the eccentricity of 1.005 just about, It's called C2012 E2 (SWAN).

Last edited by Propulsion Disk on 18 Jun 2018 07:07, edited 1 time in total.